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									Spring, 2007


                                          ISE 102


       Introduction to Linear
       Programming (LP)

       Asst. Prof. Dr. Mahmut Ali GÖKÇE
       Industrial Systems Engineering Dept.

       İzmir University of Economics
                                                                                             1
                                                                   www.izmirekonomi.edu.tr
 Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
Spring, 2007



  Introduction to Linear Programming
       Many managerial decisions involve trying to
        make the most effective use of an organization’s
        resources. Resources typically include:
           Machinery/equipment
           Labor
           Money
           Time
           Energy
           Raw materials
       These resources may be used to produce
           Products (machines, furniture, food, or clothing)
           Services (airline schedules, advertising policies, or
            investment decisions)
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Spring, 2007



  What is Linear Programming?

    Linear Programming is a mathematical technique
     designed to help managers plan and make
     necessary decisions to allocate resources
    Linear Programming (LP) is one the most widely
     used decision tools of Operations Research &
     Management Science (ORMS)
    In a survey of Fortune 500 corporations, 85 % of
     those responding said that they had used LP


Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
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Spring, 2007



  Brief History of LP


    LP was developed to solve military logistics
     problems during World War II
    In 1947, George Dantzig developed a
     solution procedure for solving linear
     programming problems (Simplex Method)
    This method turned out to be so efficient for
     solving large problems quickly
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  History of LP (contd)

   The simultaneous development of the
    computer technology established LP as an
    important tool in various fields
   Simplex Method is still the most important
    solution method for LP problems
   In recent years, a more efficient method for
    extremely large problems has been
    developed (Karmarkar’s Algorithm)
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  LP Problems

    A large number of real problems can be
     formulated and solved using LP. A partial list
     includes:
       Scheduling of personnel
       Production planning and inventory control
       Assignment problems
       Several varieties of blending problems including
        ice cream, steel making, crude oil processing
       Distribution and logistics problems
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Spring, 2007



  Typical Applications of LP

    Aggregate Planning
       Develop a production schedule which
        satisfies specified sales demands in future periods
        satisfies limitations on production capacity
        minimizes total production/inventory costs

    Scheduling Problem
       Produce a workforce schedule which
        satisfies minimum staffing requirements
        utilizes reasonable shifts for the workers
        is least costly

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Spring, 2007



  Typical Applications of LP (contd)

   Product Mix (“Blending”) Problem
      Develop the product mix which
       satisfies restrictions/requirements for customers
       does not exceed capacity and resource constraints
       results in highest profit


   Logistics
      Determine a distribution system which
       meets customer demand
       minimizes transportation costs

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Spring, 2007


               Typical Applications of LP (contd)
     Marketing
        Determine the media mix which
        meets a fixed budget
        maximizes advertising effectiveness

     Financial Planning
        Establish an investment portfolio which
        meets the total investment amount
        meets any maximum/minimum restrictions of
          investing in the available alternatives
        maximizes ROI
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Spring, 2007


               Typical Applications of LP (contd)


    What do these applications have in
     common?
     All are concerned with maximizing or
       minimizing some quantity, called the
       objective of the problem
     All have constraints which limit the
       degree to which the objective function
       can be pursued
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Spring, 2007


               Typical Applications of LP (contd)
   Fleet Assignment at Delta Air Lines
    Delta Air Lines flies over 2500 domestic flight legs
     every day, using about 450 aircrafts from 10 different
     fleets that vary by speed, capacity, amount of noise
     generated, etc.
    The fleet assignment problem is to match aircrafts
     (e.g. Boeing 747, 757, DC-10, or MD80) to flight legs
     so that seats are filled with paying passengers
    Delta is one the first airlines to solve to completion
     this fleet assignment problem, one of the largest and
     most difficult problems in airline industry
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
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Spring, 2007


               Fleet Assignment at Delta (contd)

   An airline seat is the most perishable
    commodity in the world
   Each time an aircraft takes off with an empty
    seat, a revenue opportunity is lost forever
   The flight schedule must be designed to
    capture as much business as possible,
    maximizing revenues with as little direct
    operating cost as possible

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Spring, 2007


               Fleet Assignment at Delta (contd)

     The airline industry combines
        the capital-intensive quality of the
         manufacturing sector
        low profit margin quality of the retail sector
     Airlines are capital, fuel, and labor
      intensive
     Survival and success depend on the ability
      to operate flights along the schedule as
      efficiently as possible
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
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Spring, 2007


               Fleet Assignment at Delta (contd)

    Both the size of the fleet and the number of
     different types of aircrafts have significant
     impact on schedule planning
    If the airline assigns too small a plane to a
     particular market:
     it will lose potential passengers
    If it assigns too large a plane:
     it will suffer the expense of the larger
     plane transporting empty seats
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  Stating the LP Model


   Delta implemented a large scale linear
    program to assign fleet types to flight
    legs so as to minimize a combination of
    operating and passenger “spill” costs,
    subject to a variety of operation
    constraints

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  What are the constraints?
 Some of the complicating factors include:
  number of aircrafts available in each fleet
  planning for scheduled maintenance (which
   city is the best to do the maintenance?)
  matching which crews have the skills to fly
   which aircrafts
  providing sufficient opportunity for crew rest
   time
  range and speed capability of the aircraft
  airport restrictions (noise levels!)
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Spring, 2007



  The result?!


   The typical size of the LP model that Delta
    has to optimize daily is:
      40,000 constraints
      60,000 decision variables

   The use of the LP model was expected to
    save Delta $300 million over the 3 years
    (1997)

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  Formulating LP Models
    An LP model is a model that seeks to
     maximize or minimize a linear objective
     function subject to a set of constraints

    An LP model consists of three parts:
     a well-defined set of decision variables
     an overall objective to be maximized or
      minimized
     a set of constraints
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  PetCare Problem
     PetCare specializes in high quality care for large
     dogs. Part of this care includes the assurance that
     each dog receives a minimum recommended
     amount of protein and fat on a daily basis. Two
     different ingredients, Mix 1 and Mix 2, are
     combined to create the proper diet for a dog. Each
     kg of Mix 1 provides 300 gr of protein, 200 gr of
     fat, and costs $.80, while each kg of Mix 2
     provides 200 gr of protein, 400 gr of fat, and costs
     $.60. If PetCare has a dog that requires at least
     1100 gr of protein and 1000 gr of fat, how many
     kgs of each mix should be combined to meet the
     nutritional requirements at a minimum cost?
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
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Spring, 2007



  LP Formulation Steps


           STEP 1: Understand the Problem

           STEP 2: Identify the decision variables

           STEP 3: State the objective function

           STEP 4: State the constraints


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Spring, 2007



  PetCare Problem
    PetCare specializes in high quality care for large
    dogs. Part of this care includes the assurance that
    each dog receives a minimum recommended
    amount of protein and fat on a daily basis. Two
    different ingredients, Mix 1 and Mix 2, are
    combined to create the proper diet for a dog. Each
    kg of Mix 1 provides 300 gr of protein, 200 gr of
    fat, and costs $.80, while each kg of Mix 2
    provides 200 gr of protein, 400 gr of fat, and
    costs $.60. If PetCare has a dog that requires at
    least 1100 gr of protein and 1000 gr of fat, how
    many kgs of each mix should be combined to meet
    the nutritional requirements at a minimum cost?
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
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Spring, 2007



  PetCare: LP Formulation
    STEP 1: Understand the Problem
    STEP 2: Identify the decision variables
        x1 : kgs of mix 1 to be used to feed the dog
        x2 : kgs of mix 2 to be used to feed the dog
    STEP 3: State the objective function
        minimize        0.8 x1 + 0.6 x2                           (total cost)
    STEP 4: State the constraints
               subject to 300 x1 + 200 x2  1100                  (protein constraint)
                          200 x1 + 400 x2  1000                  (fat constraint)
                                       x1  0                     (sign restriction)
                                       x2  0                     (sign restriction)


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Spring, 2007



  Furnco Company Problem

    Furnco manufactures desks and chairs. Each
    desk uses 4 units of wood, and each chair uses 3
    units of wood. A desk contributes $40 to profit,
    and a chair contributes $25. Marketing restrictions
    require that the number of chairs produced must
    be at least twice the number of desks produced.
    There are 20 units of wood available. Formulate
    the Linear Programming model to maximize
    Furnco’s profit.

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  Furnco Company (contd)
      x1 : number of desks produced
      x2 : number of chairs produced

      maximize 40 x1 + 25 x2                                      (objective
       function)

      subject to              4 x1 + 3 x2  20 (wood constraint)
                              2 x1 - x2  0 (marketing constraint)
                               x1 , x2  0 (sign restrictions)

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  Farmer Jane Problem
         Farmer Jane owns 45 acres of land. She is going
         to plant each acre with wheat or corn. Each acre
         planted with wheat yields $200 profit; each with
         corn yields $300 profit. The labor and fertilizer
         used for each acre are as follows:
                         Wheat         Corn
             Labor       3 workers 2 workers
             Fertilizer 2 tons         4 tons
         100 workers and 120 tons of fertilizer are
         available. Formulate the Linear Programming
         model to maximize the farmer’s profit.

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  Farmer Jane (contd)

   x1 : acres of land planted with wheat
   x2 : acres of land planted with corn

   maximize                200 x1 + 300 x2                        (objective function)

   subject to                  x1 +        x2  45                (land constraint)
                           3 x1 + 2 x2  100 (labor constraint)
                           2 x1 + 4 x2  120 (fertilizer constraint )
                                   x1 , x2  0                          (sign restrictions)
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  Truck-Co Company Problem
         Truck-co manufactures two types of trucks: 1 and
         2. Each truck must go through the painting shop
         and the assembly shop. If the painting shop were
         completely devoted to painting type 1 trucks, 800
         per day could be painted, whereas if it were
         completely devoted to painting type 2 trucks, 700
         per day could be painted. Is the assembly shop
         were completely devoted to assembling truck 1
         engines, 1500 per day could be assembled, and if
         it were completely devoted to assembling truck 2
         engines, 1200 per day could be assembled. Each
         type 1 truck contributes $300 to profit; each type
         2 truck contributes $500. Formulate the LP
         problem to maximize Truckco’s profit.

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  Truckco Company (contd)

x1 : number of type 1 trucks manufactured
x2 : number of type 2 trucks manufactured

maximize              300 x1 + 500 x2                      (objective function)

subject to            7 x1 + 8 x2  5600                          (painting constraint)
                   12 x1 + 15 x2  18000 (assembly constraint)
                                x1 , x2         0         (sign restrictions)

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  McDamat’s Fast Food Restaurant
     McDamat's fast food restaurant requires different number of full
     time employees on different days of the week. The table below
     shows the minimum requirements per day of a typical week:
      Day of week      Empl Reqd      Day of week     Empl Reqd
       Monday              7          Friday                 4
       Tuesday             3          Saturday               6
       Wednesday           5          Sunday                 4
       Thursday            9
     Union rules state that each full-time employee must work 5
     consecutive days and then receive 2 days off. The restaurant wants
     to meet its daily requirements using only full time personnel.
     Formulate the LP model to minimize the number of full time
     employees required.

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  McDamat’s Fast Food Restaurant
  (contd)
     Defining Decision Variables
         xi : number of employees beginning
      work on day i           where i = Monday,
      …. , Sunday

     Defining the Objective Function
          min Z = xmon + xtue + xwed + xthu + xfri +
      xsat + xsun
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McDamat’s Fast Food Restaurant (contd)
 Defining the Constraint Set
      xmon + xthu + xfri + xsat + xsun  7 (Mon Reqts)
      xmon + xtue + xfri + xsat + xsun  3 (Tue Reqts)
      xmon + xtue + xwed + xsat + xsun  5(Wed Reqts)
      xmon + xtue + xwed + xthu + xsun  9(Thu Reqts)
      xmon + xtue + xwed + xthu + xfri  4       (Fri Reqts)
      xtue + xwed + xthu + xfri + xsat  6 (Sat Reqts)
      xwed + xthu + xfri + xsat + xsun  4 (Sun Reqts)
 Non-Negativity Condition (Sign Restriction)
     xmon , …. , xsun  0
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A Multi-Period Production Planning Pr.
   Sailco Corporation must determine how many sailboats to produce
   during each of the next four quarters. The demand during each of
   the next four quarters is as follows:
        Quarters            1         2      3       4 .
        Demand                   40      60    75    25
   At the beginning of the first quarter Sailco has an inventory of 10
   sailboats.
   At the beginning of each quarter Sailco must decide how many
   sailboats to produce that quarter. Sailboats produced during a
   quarter can be used to meet demand for that quarter.
                               Capacity        Cost         .
        Regular Time           40 (sailboats) $400/sailboat
        Overtime                               $450/sailboat
   Inventory Holding Cost: $20/sailboat
   Determine a production schedule to minimize the sum of production
   and inventory holding costs during the next four quarters.
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  A Multiperiod PP Problem (contd)
   Defining Decision Variables
         R1 : regular time production at quarter 1
         R2 : regular time production at quarter 2
         …
         …
         Rt : regular time production at quarter t
         Ot : overtime production at quarter t
         It : inventory at the end of quarter t

   Defining the Objective Function
    min 400 R1 + 400 R2 + 400 R3 + 400 R4 + 450 O1 + 450 O2 +
         450 O3 + 450 O4 + 20 I1 + 20 I2 + 20 I3 + 20 I4



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  A Multiperiod PP Problem (contd)
   Defining the Constraint Set
         10 + R1 + O1 - I1 = 40
         I1 + R2 + O2 - I2 = 60
         I2 + R3 + O3 - I3 = 75
         I3 + R4 + O4 - I4 = 25
                          R1  40
                          R2  40
                          R3  40
                          R4  40

   Non-Negativity Condition (Sign Restriction)
        R1, R2, R3, R4, O1, O2, O3, O4, I1, I2, I3, I4  0


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  LP Summary

   An LP problem is an optimization problem for
    which we do the following:
       We attempt to maximize (or minimize) a linear function
        of the decision variables. The function that is to be
        maximized (or minimized) is called the objective
        function
       The values of the decision variables must satisfy a set
        of constraints. Each constraint must be a linear
        equation or linear inequality
       A sign restriction is associated with each variable

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  Graphical Solution Method
Spring, 2007




        X2
     Chairs

          7                                                       Furnco Company
                                               (2)
 [166.75] 6.67                                                    Max     40 x1 + 25 x2
         6                                                        s.t.            4 x1 + 3
                                                                      x2  20
         5                                                                2 x1 - x2  0
         4
                                                                          x1 , x2  0
                                       (2,4) [180]



         3


         2


         1


          0                    2.5       3.754
                         2                             5    6      7     X1
                                                                         Desks
                                                           (1)
                                               Z=150
                                     Z=100

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  Graphical Solution Method (contd)
      X2            (4)
   Corn

      60
                                                                  Farner Jane (modified)
                                                                  max    200 x1 + 300 x2
      50
                                                                  s.t      x1 + x2  45
        45
                                                                         3 x1 + 2 x2  100
      40                                                               2 x1 + 4 x2  120
                                   (20,20)
                                                                           x1          ≥ 10
      30                                                                    x1 , x 2  0
               (10,25)

       20                                               (30,15)



      10
                                                                       (3)
                                               (2)        (1)
           0                              33.3
                              20                       40 45      50              X1
                         10




                                                                            60
                                     30




                                                                                  Wheat
                  [2000]                     [6667]
                                                                   Z=7080
                                                      Z=6000
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  Special Cases of the Feasible Region




                  Infeasible                                      Redundant Constraint

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           More Special Cases of the Feasible
           Region




    Unbounded Feasible Region                                Unbounded Feasible Region
       Unbounded Solution                                    Bounded Optimal Solution
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  Special Cases of the Optimal Solution




               Multiple Optima                                    Unbounded Solution

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